Theorem aaanv 45027

Description: Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2371. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
aaanv	⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem aaanv

Step	Hyp	Ref	Expression
1		nfv 1941	. . 3 ⊢ Ⅎ𝑦𝜑
2		nfv 1941	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	aaan 2371	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
4	3	bicomi 227	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-11 2198  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811

This theorem is referenced by: (None)